INTRODUCTION TO STRONG FIELD QED

INTRODUCTION TO STRONG
FIELD QED
TOM HEINZL
GK SEMINAR, TPI JENA
21/04/2009
with: C. Harvey (UoP), A. Ilderton (Dublin), K. Ledingham (Strathclyde), H. Schwoerer
(Stellenbosch), R. Sauerbrey and U. Schramm (FZD), A. Wipf (FSU Jena)

Outline
1. Introduction
2. Strong Field Electrodynamics
Application I: Relativistic charges in laser fields
3. Quantum Electrodynamics
4. Strong Field QED
Application II: vacuum polarisation effects
Application III: nonlinear Compton scattering [tomorrow]
5. Summary and Conclusions

1. Introduction

Explaining the title
“QED” = quantum electrodynamics
Quantum gauge field theory
describes interactions of “light” and “matter”
“light”: photons
“matter”: charged elementary particles
Leptons: electron, muon, tau
Quarks: up, down, strange, … (bound into hadrons)
NB1: mainly discuss electrons
NB2: “matter” includes anti-matter (positrons etc.)

Explaining the title cont d
“Strong Field”
Throughout this lecture: strong field = laser
Typical magnitudes assumed are
Power
Intensity
Electric field
Magnetic field

Strong field parameter
‘dimensionless laser amplitude’
(purely classical) ratio (no ):
NB: implies relativistic quiver motion of
electrons in laser beam

2. Strong Field Electrodynamics
Relativistic charges in electromagnetic fields

Motivation
Direct laser acceleration (DLA) in vacuum –
Usual RF cavities break down at ‘critical’ electric field
of about 10 8 V/m
Can one use a laser to accelerate particles?
No – in a perfect plane wave (Lawson-Woodward theorem)
Yes – in ‘realistic’ laser fields
experiments at proof-of-principle stage
30 → 30.03 MeV (Plettner, PRL 95, 134801 (2005))
40 → 43.7 MeV (Campbell et al., IEEE TPS 28, 1094 (2000))

Covariant equation of motion
Charge e, mass m in field
Equation of motion:
where dot denotes derivative w.r.t. proper time
R.h.s.: relativistic Lorentz force
Task: find trajectory
NB: as EoM in general nonlinear!

Simplification I: constant fields
EoM becomes linear for constant fields,
Write in matrix form
where
First integral :
Second integral: 4 cases – depending on invariants

Constant fields: 4 cases (Taub 1948)
Table:
Name
Loxodromic
Elliptic
Hyperbolic
Parabolic
Field configuration
(special frame)
Invariant characterisation
(frame independent)
NB: parabolic = crossed or null fields,

4 cases: illustration
Hyperbolic
Loxodromic
(C. Harvey)
Elliptic
Parabolic
NB: net acceleration
Hyperbolic: basic
principle of standard
accelerator
Loxodromic:
accelerator with
magnetic focussing
Parabolic: laser
subcycle acceleration
Elliptic: no
acceleration

Simplification II: plane waves
Covariant description of plane wave field
Light-like wave 4-vector , →
dispersion for massless particles
Ingredients
Invariant phase
Field strength:
Transversality:
Null field:
Conservation law:
makes EoM linear, hence integrable

Illustration I: linear polarisation
Trajectory of charge in average rest frame
(mean drift velocity subtracted)
Lissajous 2:1

Illustration II: Circular polarisation
Trajectory of charge in average rest frame
(mean drift velocity subtracted)

DLA: Conclusion
Motion in periodic plane wave fields ( )
Periodic and bounded
Hence no net acceleration (Lawson-Woodward
theorem)
Loophole: give up periodicity, e.g. crossed fields
( )
Motion aperiodic and unbounded
Net (‘subcycle’) acceleration

DLA: Outlook
Work in progress
realistic laser fields
include radiation loss
numerical scheme
Application
Plug calculated orbits into Larmor formula
Determine classical radiation spectrum
(Sarachik/Schappert 1970, Esarey et al. 1996, ...)

3. QED
3.1 Introduction

Switching on h-bar
Recall dimensionless laser amplitude: energy gain
across laser wavelength in units of
for relativistic electrons
Replace laser by Compton wavelength
and demand energy gain across to be

Critical electric field
This yields ‘critical’ electric field (Sauter 1931)
In this field a change in energy occurs within
microscopic length scale
Hence, it becomes possible to create electron
positron pairs from vacuum
Presence of and c: relativity ∪ quantum theory
Need relativistic quantum field theory: QED!

QED Lagrangian
Compact version:
Covariant derivative
guarantees gauge invariance under
→ photon massless
determines interaction

QED: basic interaction
Rewrite interaction term
So, coupling of gauge field to Dirac current
Pictorially: Feynman diagram of ‘QED vertex’
Dirac field
Photon field
coupling strength
Dirac field

Feynman diagrams
Tree level, e.g.
Loops, e.g.
Compton scattering
O( )
Typically finite
[tomorrow’s talk]
Vacuum polarisation
O( )
Typically infinite
If so, renormalise

Photon-photon scattering
Loop effects imply photon-photon coupling mediated
by virtual Dirac particles
Feynman diagram (finite due to gauge invariance!)
(Low energy: Euler, Heisenberg,
Kockel 1935, 1936
High energy: Akhiezer 1937)
Induced nonlinearity: effective terms in
Lagrangian, in EoM

Photon-photon scattering cont d
Low energy analysis:
At low energy: virtual loop not resolved
Obtain effective theory with effective vertices
Heisenberg-Euler effective Lagrangian: nonlinear
quantum correction to Maxwell theory

Heisenberg-Euler
Discussion:
from 4 QED vertices
from dimensional analysis (Lagrangian has mass
dimension 4 in d=4)
and from gauge invariance
recall: ,
Coefficients and from detailed calculation

Photon-photon scattering cont d
Low-energy cross section
Invariant amplitude: each gradient from etc.
produces frequency factor
2
X-section (optical regime):
NB: not measured yet (→ bounds)
suggestion for Astra Gemini: (Marklund et al., 2006)

Loops vs. trees: optical theorem
Optical theorem (Kramers-Kronig):
“the imaginary part of the forward scattering
amplitude is proportional to the total cross section”
Example: photon-photon scattering
Scattering loop
Pair production cross
section → absorption

4. Strong Field QED
Vacuum polarisation effects

Strong field QED
Recall QED interaction
Assume presence of external strong laser field
Get additional interaction (no )
Include into free Lagrangian
Main effect: replace free Dirac electrons by Volkov
electrons (electrons ‘dressed’ by e.m. wave)
Pictorially: ‘dressed’ (Volkov) electron line

Field decomposition
Two types of photons
Weak ‘probe field’ : perturbative
Strong background field : nonperturbative
Effective Lagrangian = expansion in
Goal: determine ‘coefficients’ exactly
Only possible for special backgrounds

Strong field vacuum polarisation
Consider leading (bilinear) order
coefficient = dressed vacuum polarisation loop:
Polarisation
tensor
QED
LO SFQED

Polarisation tensor: crossed fields
Simplest case: crossed fields (constant null fields)
Consequence: laser background has
Two remaining invariants:
Probe 4-momentum squared:
where is index of refraction due to laser BG
Energy density seen by probe:
BG energy-momentum tensor

crossed fields cont d
Determine relevant eigenvalues of
: two nontrivial dispersion relations
NB: can be viewed as stemming from effective
metrics
Result:
‘vacuum’ birefringence

Indices of refraction
Formula (Toll 1952)
Two small parameters
dimensionless field strength (recall: )
dimensionless probe frequency
Note dependence on product

Experiment: measure ellipticity
Phase retardation of e +

Analysis
ellipticity (squared)
Power law suppressed…
Optimal scenario @ ELI
large intensity:
large probe frequency (X-ray, ):
Still experimental challenge!

New frontiers: increase parameters
Theory: for large , refractive indices develop
imaginary part, (Toll 1952)
Reason: pair creation (→ optical theorem)
Q: can one get there?
Large : currently science fiction
Large : use Compton backscattering!

Parameter range
: Compton backscattering off high-energy
(from linac or wake field accelerator)
SLAC
10 6 Vulcan
10 3
ELI
Backscattered
(5 GeV )
1
1 PW
10 PW
all-optical
10 -6 10 -4 10 -2 1

Large-ν birefringence I
for : 3 GeV @ ELI, 10 GeV @ Vulcan10PW
Toll 1952
Shore 2007
NB: SLAC E-144 had
(K. Langfeld)

Large-ν birefringence II
For , : find anomalous dispersion and
Possibly, alternative signal for PP
Subtle interplay between probe energy (ν) and laser
intensity (ǫ 2 )
Open questions:
Polarimetry for high-energy γ’s ?
Experimental signatures ?

5. Conclusion

Strong laser fields
Direct laser acceleration in vacuum
Proof of principle
More advanced: plasma wake field acceleration
Strong field QED
Absorptive: Pair creation – at which field strength?
Dispersive: vacuum birefringence
Scattering processes: no thresholds
Gamma-gamma scattering
High-intensity Compton